Video Transcript
Find the first derivative of the
function 𝑦 equals negative seven 𝑥 to the fourth power times the natural log of
six 𝑥 to the fourth power.
Here, we have the product of two
differentiable functions. The first is negative seven 𝑥 to
the fourth power and the second is the natural log of six 𝑥 of the fourth
power. We can, therefore, find the first
derivative of our function by using the product rule. This says that the derivative of
the product of two differentiable functions, 𝑢 and 𝑣, is 𝑢 times d𝑣 by d𝑥 plus
𝑣 times d𝑢 by d𝑥. If we let 𝑢 be equal to negative
seven 𝑥 to the fourth power, then d𝑢 by d𝑥 is equal to four times negative seven
𝑥 cubed. That’s negative 28𝑥 cubed. We then let 𝑣 be equal to the
natural log of six 𝑥 to the fourth power.
So how do we obtain d𝑣 by d𝑥? Well, we can do one of two
things. We could spot that we have a
composite function and use the chain rule to find its derivative. Alternatively, we can quote the
general result for the derivative of the logarithm of some function 𝑓 of 𝑥. This says that if 𝑦 is equal to
the natural log of this function 𝑓 of 𝑥, then d𝑦 by d𝑥 is equal to 𝑓 prime of
𝑥, the derivative of that function, divided by the original function 𝑓 of 𝑥. In this case, 𝑓 of 𝑥 is equal to
six 𝑥 to the fourth power. So 𝑓 prime of 𝑥, the derivative
of this, is four times six 𝑥 cubed, which is 24𝑥 cubed. d𝑣 by d𝑥 is, therefore,
24𝑥 cubed, divided by the original function, six 𝑥 to the fourth power.
Well, this simplifies quite nicely
to four over 𝑥. And we can now substitute
everything we have into the formula for the product rule. d𝑦 by d𝑥 is 𝑢 times d𝑣 by d𝑥. That’s negative seven 𝑥 to the
fourth power times four over 𝑥 plus 𝑣 times d𝑢 by d𝑥. That’s the natural log of six 𝑥 to
the fourth power times negative 28𝑥 cubed. We simplify, and then we factor
negative 28𝑥 cubed. And we obtained d𝑦 by d𝑥 to be
negative 28𝑥 cubed times the natural logarithm six 𝑥 to the fourth power plus
one.
